Homoclinism of groups

Definition

For any group $G$, let $\operatorname {Inn} (G)$ denote the inner automorphism group of $G$, $G'$ denote the derived subgroup of $G$, and $Z(G)$ denote the center of $G$.

Let $\gamma _{G}$ denote the map from $\operatorname {Inn} (G)\times \operatorname {Inn} (G)$ to $G'$ defined by first taking the map $G\times G\to G'$ given as $(x,y)\mapsto x^{-1}y^{-1}xy$ and then observing that the map is constant on the cosets of $Z(G)\times Z(G)$.

A homoclinism of groups $G_{1}$ and $G_{2}$ is a pair $(\zeta ,\varphi )$ where $\zeta$ is a homomorphism from $\operatorname {Inn} (G_{1})$ to $\operatorname {Inn} (G_{2})$ and $\varphi$ is a homomorphism from $G_{1}'$ to $G_{2}'$ such that $\varphi \circ \gamma _{G_{1}}=\gamma _{G_{2}}\circ (\zeta \times \zeta )$. In symbols, this means that for any $x,y\in \operatorname {Inn} (G_{1})$ (possibly equal, possibly distinct), we have:

$\varphi (\gamma _{G_{1}}(x,y))=\gamma _{G_{2}}(\zeta (x),\zeta (y))$

Pictorially, the following diagram must commute:

${\begin{array}{ccc}\operatorname {Inn} (G_{1})\times \operatorname {Inn} (G_{1})&{\stackrel {\zeta \times \zeta }{\to }}&\operatorname {Inn} (G_{2})\times \operatorname {Inn} (G_{2})\\\downarrow ^{\gamma _{G_{1}}}&&\downarrow ^{\gamma _{G_{2}}}\\G_{1}'&{\stackrel {\varphi }{\to }}&G_{2}'\\\end{array}}$

Related notions

• Category of groups with homoclinisms is an interesting category that differs somewhat from the usual category of groups.
• Isoclinism is an invertible homoclinism.